Session LM: Turbulent Mixing II

Joint Conserved Scalar Statistics in Three-stream Mixing

The double scalar mixing layer (DSML) has been studied recently as a model problem for a piloted jet flame, which is a special case of three-stream mixing of particular interest to combustion researchers. In this special case, mixing can be described in terms of a single conserved scalar. In the more general case, where the inner stream is not a mixture of the two outer streams, two conserved scalars are necessary, one to represent mixing of each of the outer streams (as in binary mixing) and one to represent mixing of the inner stream. We present and compare joint statistics obtained for these two conserved scalars using DNS and a Lagrangian stochastic model. Because of the extra dimension in concentration space, sampling is more difficult than for the DSML, but as for the DSML we find good agreement between the DNS and model results for a range of unconditional and conditional scalar statistics, including joint moments and the joint pdf, the mean velocity conditioned on both scalars and the conditional diffusion.   

Multi-scale laminar flows with turbulent-like properties

By applying fractal electromagnetic force fields on a thin layer of brine, we generate steady quasi-two-dimensional laminar flows with multi-scale stagnation point topology. This topology is shown to control the evolution of pair separation ($\Delta$) statistics by imposing a turbulent-like locality based on the sizes and strain rates of hyperbolic stagnation points when the flows are fast enough, in which case $<\Delta^{2}> \sim t^{\gamma}$ (whether $t$ stands for time) is a good approximation with $\gamma$ close to 3. Spatially multi-scale laminar flows with turbulent-like spectral and stirring properties are a new concept with potential applications in efficient and micro-fluidic mixing.   

Is a randomly stirred mixture a maximum entropy object?

A randomly stirred scalar mixture can be viewed as a set of adjacent stretched sheets, merging diffusively into each other. For a mixture decaying at fixed stirring intensity and average concentration, the consequences of this vision offer a precise description of the shape and evolution of the concentration Probability Density Function [PRL 91 (18), 184501, (2003)]. Because of the aggregation between nearby sheets, the concentration field is defined on a naturally coarse grained scale $\eta$ incorporating many independent merged sheets. There is obviously a number of ways to realize a given macroscopic concentration distribution by assembling elementary sheets into independent boxes of size $\eta$. From the inventory of the number of microscopic states, we derive the corresponding entropy, and show that real mixtures {\it do~not} maximize entropy. Real and maximum entropy distributions are however close in shape and evolution, and both consistently lead to the same value of the entropy at complete mixing.   

Using the theory of shear dispersion to estimate river mixing

We evaluate the use of river velocity and bathymetry data measured with an acoustic Doppler current profiler (ADCP) to estimate the longitudinal dispersion coefficient $K$. If shear dispersion controls the mixing, the dispersion coefficient can be estimated from a theoretical formula involving velocity measurements in a cross section. The relative ease and detail with which ADCPs measure velocities allows the dispersion coefficient to be estimated with greater frequency and spatial coverage in U.S. rivers. Comparing values of $K$ computed from ADCP measurements from the U.S. Geological Survey with values derived from tracer studies shows that the ADCP works as well as or better than empirical formulas for the dispersion coefficient. A possible source of discrepancy in this comparison is that the ADCP measurements and tracer measurements were conducted many years apart. To address this issue, we conducted simultaneous ADCP measurements and tracer studies in two Florida rivers. In both rivers, the ADCP method underestimated the dispersion coefficient, but compared to empirical formulas, the ADCP method provides the most reliable estimate.   

Three-dimensional measurements of scalar dispersion in grid turbulence

The three-dimensional scalar-dispersion field of a passive Lagrangian-marker contaminant released in grid-generated turbulent flow with an initial Taylor Reynolds number of $\sim 40$, was investigated. The experiments were conducted in the GALCIT Free Surface Water Tunnel using laser-induced fluorescence, a two-dimensional galvanometric optical scanner, and recorded with a fast-framing CCD camera and data acquisition system that sustained $10^8\,$pix/s, at $12\,$bits/pix. The resulting data frames were processed to compensate for temporal and spatial skewing, and resampled on a Cartesian grid for subsequent processing. The spatial structure of the scalar field downstream of the grid will be presented and discussed.   

Scalar Dissipation Length Scale Measurement

In the current study, mixture fraction in non-reacting turbulent C2H4/air circular jet flows at Reynolds number of 4,000, 6,000, 10,000 and 15,200, measured with 1D line laser Rayleigh scattering technique, was used to determine the dissipation length scale and corresponding spatial resolution required to resolve this scale. The dissipation length scale is experimentally determined from both the measured 1D dissipation spectrum and spatial filtering. The dissipation length scale is found to be $\sim $6 times larger than the Batchelor scale, which is consistent with the Batchelor frequency in time series measurement and strain limited mass diffusion length scale in 2D imaging experiments. The spatial resolution to resolve this physical limiting dissipation length scale is not simply related to the Nyquist criteria and a systematic approach is proposed by considering both experimental and numerical factors. These techniques can be extended to complex turbulent flows, e.g. turbulent reacting flows, where isotropic turbulent theory may not apply. A new form of the 1D dissipation model spectrum is developed based on the dissipation structures and compared with Pope's model spectrum.   

Numerical analysis of scalar dissipation length-scales and their scaling properties

Scalar dissipation rate, $\chi $, is fundamental to the description of scalar-mixing in turbulent non-premixed combustion. Most contributions to the statistics for $\chi $ come from the finest turbulent mixing-scales and thus its adequate characterisation requires good resolution. Reliable $\chi $-measurement is complicated by the trade-off between higher resolution and greater signal-to-noise ratio. Thus, the present numerical study utilises the error-free mixture fraction, Z, and fluid mechanical data from the turbulent reacting jet DNS of Pantano (2004). The aim is to quantify the resolution requirements for $\chi $-measurement in terms of easily measurable properties of the flow like the integral-scale Reynolds number, $Re_\delta $, using spectral and spatial-filtering [cf. Barlow and Karpetis (2005)] analyses. Analysis of the 1-D cross-stream dissipation spectra enables the estimation of the dissipation length scales. It is shown that these spectrally-computed scales follow the expected Kolmogorov scaling with $Re_\delta ^{-0.75} $. The work also involves local smoothening of the instantaneous $\chi $-field over a non-overlapping spatial-interval (filter-width, $w_f )$, to study the smoothened $\chi $-value as a function of $w_f $, as $w_f $ is extrapolated to the smallest scale of interest. The dissipation length-scales thus captured show a stringent $Re_\delta ^{-1} $ scaling, compared to the usual Kolmogorov-type. This concurs with the criterion of 'resolution adequacy' of the DNS, as set out by Sreenivasan (2004) using the theory of multi-fractals.   

Entrainment and mixing in a turbulent transverse jet: a DNS study

DNS of passive scalar mixing in a round turbulent transverse jet is performed at conditions matching that of experiment (Su and Mungal, {\it J. Fluid Mech.} 2004). The velocity ratio is 5.7 and the jet Reynolds number is 5000. The simulation is validated by detailed comparison of mean velocity, turbulence intensities, and scalar concentration to experiment. The simulation data is used to discuss the turbulent kinetic energy budget, and different timescales present in the flow, entrainment characteristics and mechanisms, and possible reasons why RANS computations do not predict this flow field adequately.   

Dynamics of Passive Scalar Gradients In A Turbulent Kolmogorov Flow

Using a direct numerical simulation of a low-Reynolds number Kolmogorov flow, we investigate the dynamics of passive scalar gradients in relation to the dynamics of turbulent structures. The Schmidt number ranges from 1 to 10 with full resolution of the Batchelor scale of the passive scalar. We are particularly interested in the small scale mixing occurring around vortices, in regions that are characterized by their strong anisotropy and their dominant energetic role on the dynamics of turbulence. Coherence in the formation of sharp gradients of the passive scalar concentration is studied using the second $Q$ and third $R$ invariants of the velocity gradient tensor, as well as conditional statistics based on vortices.   

Passive scalar mixing in vortex rings

Direct numerical simulations of passive scalar mixing in vortex rings are performed, with and without crossflow. The simulation results without crossflow agree well with experimental data for `formation number', total circulation, trajectory and entrainment fraction. Scalar profiles, mixedness and volume of scalar carrying fluid are used to quantify mixing, whose characteristics are quite different in the formation and propagation phases of the ring. These results are explained in terms of entrainment by the ring. The simulations with crossflow show that the ring tilts and deforms. When the stroke ratio is greater than formation number, the ring tilts in the direction of the crossflow. On the other hand, when the stroke ratio is less than formation number, the ring tilts in the opposite direction, such that its induced velocity opposes the crossflow. The Magnus effect may be used to provide a simple explanation. The impact of this behavior on mixing will be discussed.   

Experimental observation of particles in a superfluid counterflow

We introduce micron-sized solid hydrogen particles into superfluid helium and demonstrate that they adhere to the cores of quantized vortices. We then observe the particles in a thermal counterflow near 2.1 Kelvin and report several interesting phenomena that depend on the volume fraction of hydrogen in the system. At volume fractions near $10^{-4}$, the particles collect onto continuous filaments that form stable networks. At lower volume fractions near $10^{-6}$, we observe that individual particles move with two separate types of motion. In the context of the two-fluid model, we attribute one of these motions to particles moving with the normal fluid, and the other to particles confined to move along quantized vortex lines in the superfluid. Although particles are routinely used to reveal the motions of classical fluids, their role in studying flows in superfluid helium must be determined. As is well known, the response of particles to the motions of superfluid helium is often complicated and difficult to interpret, due to the particles' interaction with quantized vortices. In addition, the response of the fluid to the presence of particles is not well understood. We show that in some cases the particles cannot be considered passive tracers of the flow, even when present at low volume fractions.